integral of (x^2-15)/(x+7)

解

\int \frac{x ^{2} - 15}{x + 7} d x

\frac{1}{2} (x + 7)^{2} - 14 x - 98 + 34 ln ∣ x + 7 ∣ + C

解答ステップ

\int \frac{x ^{2} - 15}{x + 7} d x

拡張

\frac{x ^{2} - 15}{x + 7} : \frac{x ^{2}}{x + 7} - \frac{15}{x + 7}

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x ^{2}}{x + 7} d x - \int \frac{15}{x + 7} d x

\int \frac{x ^{2}}{x + 7} d x = \frac{( x + 7 ) ^{2}}{2} - 14 (x + 7) + 49 ln ∣ x + 7 ∣

\int \frac{15}{x + 7} d x = 15 ln ∣ x + 7 ∣

= \frac{( x + 7 ) ^{2}}{2} - 14 (x + 7) + 49 ln ∣ x + 7 ∣ - 15 ln ∣ x + 7 ∣

簡素化

\frac{( x + 7 ) ^{2}}{2} - 14 (x + 7) + 49 ln ∣ x + 7 ∣ - 15 ln ∣ x + 7 ∣ : \frac{1}{2} (x + 7)^{2} - 14 x - 98 + 34 ln ∣ x + 7 ∣

= \frac{1}{2} (x + 7)^{2} - 14 x - 98 + 34 ln ∣ x + 7 ∣

定数を解答に追加する

= \frac{1}{2} (x + 7)^{2} - 14 x - 98 + 34 ln ∣ x + 7 ∣ + C